The basic task in this chapter is, given an “expression” f, say f ∈ F(x), where F is a field, to compute the indefinite integral ∫ f = ∫ f(x)dx, that is, another “expression” (possibly in a larger domain) g with g′ = f, where ′ denotes differentiation with respect to the variable x. “Expressions” are usually built from rational functions and “elementary functions” such as sin, cos, exp, log, etc. (Since it is more common, we denote the natural (base e) logarithm by “log” instead of “ln” in this chapter.) Such integrals need not exist: Liouville's (1835) theorem implies that exp(x2) has no integral involving only rational functions, sin, cos, exp, and log.

A practical approach to the symbolic integration problem is to use a plethora of formulas for special functions, tricks from basic calculus like substitutions and integration by parts, and table lookups. There are projects that load the whole contents of existing printed integral tables into computer algebra systems, using optical character recognition, and modern computer algebra systems can solve practically all integration exercises in calculus textbooks. In the following, we discuss a systematic algorithm in the case of rational and “hyperexponential” functions as integrands. This approach can be extended—with plenty of new ideas and techniques—to more general functions, but we do not pursue this topic further.